Such an arrangement is known from German Application DE-A 42 22 227 and is capable of identifying given substances, notably crystalline substances (explosives), in a piece of luggage on the basis of their pulse transfer spectrum. The polychromatic X-rays generated by a single X-ray source are shaped so as to form a conical primary beam by means of a slit aperture in a primary diaphragm plate. The radiation scattered in the examination zone is incident, via slits of a secondary diaphragm device which may consist of one or more diaphragm plates, on a detector consisting of a plurality of elements. In the case of a half angle of aperture of the primary beam of 0.0309 radians, a distance of 1,187 mm between the primary diaphragm plate and the radiation source, and a thickness of the examination zone of 500 mm in the radiation direction, the diameter of the primary radiation beam at approximately the center of the examination zone (i.e. at a distance of 250 mm from the primary diaphragm plate) amounts to approximately 90 mm. Because, generally speaking, the objects to be examined have substantially larger dimensions, it is necessary to perform a meander-like scanning motion during which all areas of the object are successively examined.